Question

is hijk a square? justify your answer. yes, all sides are congruent, and \\(\overline{hi}\\) and \\(\overline{jk}\\) are perpendicular. yes, all sides are congruent, and \\(\overline{hi}\\) and \\(\overline{ij}\\) are perpendicular. no, \\(\overline{hi}\\) and \\(\overline{ij}\\) are not perpendicular. no, not all sides are congruent.

Explanation:

Step1: Identify coordinates of vertices

$H(2, -5), I(6, -1), J(2, 3), K(-2, -1)$

Step2: Calculate side lengths

Use distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

• $\overline{HI}$: $\sqrt{(6-2)^2+(-1-(-5))^2}=\sqrt{16+16}=\sqrt{32}=4\sqrt{2}$
• $\overline{IJ}$: $\sqrt{(2-6)^2+(3-(-1))^2}=\sqrt{16+16}=\sqrt{32}=4\sqrt{2}$
• $\overline{JK}$: $\sqrt{(-2-2)^2+(-1-3)^2}=\sqrt{16+16}=\sqrt{32}=4\sqrt{2}$
• $\overline{KH}$: $\sqrt{(2-(-2))^2+(-5-(-1))^2}=\sqrt{16+16}=\sqrt{32}=4\sqrt{2}$

All sides are congruent.

Step3: Calculate slopes of adjacent sides

Slope formula $m=\frac{y_2-y_1}{x_2-x_1}$

• Slope of $\overline{HI}$: $\frac{-1-(-5)}{6-2}=\frac{4}{4}=1$
• Slope of $\overline{IJ}$: $\frac{3-(-1)}{2-6}=\frac{4}{-4}=-1$

Step4: Check perpendicularity

Product of slopes: $1\times(-1)=-1$, so $\overline{HI}\perp\overline{IJ}$.
A quadrilateral with 4 congruent sides and perpendicular adjacent sides is a square.

Answer:

Yes, all sides are congruent, and $\overline{HI}$ and $\overline{IJ}$ are perpendicular.